We will use what is usually called the "shift theorem" from transform theory 

 which states that, if G(f ) is the Fourier transform of g(x) then, from equation 

 (24), e" l2lrf x A G(f ) is the Fourier transform of g(x-A). With A = Sy, this 

 theorem is used to integrate out the x dependence in equation (25) resulting in 

 the equation 



F(f,f) = f G(f)e- i2lI >' (f x S+f y ) dy (26) 



v x y J x 



7 -y 



and, by symmetry, 



(27) 



F(f x ,f y ) = 2G(y / cos [2ir/(f x S+f y )] dy. 



Integrating over y yields 



Sin [2TrY(f S +f )] 



F(f ,f ) = 2G(f ) L c+ f / ° r 



xy x 2ir(fb+tj 



' x y 



F(f ,f ) = 2YG(f ) Sine [2ttY (f S + f )j . (28) 



x y x L x y J 



Since the Sine function has its maximum value when the argument is zero, 



equation (28) defines a function in the (f ,f ) plane possessing a trend with 



-f X y 



S =-£ . Thus we see that, since trends in the (x,y) plane are mapped into 



x 

 trends in the (f ,f ) plane in the orthogonal direction, survey track direction 



x y 



may be determined from the two-dimensional FFT by aligning the tracks parallel 



to the trends in the (f ,f ) plane. With this direction specified, the shape of 

 x y 



the two-dimensional amplitude spectrum is used in conjunction with the Shannon 

 sampling theorem to estimate the band limits of g(x,y) which, in turn, define the 

 required track spacing and sampling rate for the survey. A general two-dimensional 

 FFT computer program utilizing equally spaced gridded data and a fast Fourier 



17 



